# Revolution of two curves around the y-axis

I've looked through the other posts on rotating a function around the x or y axis and haven't had any luck.

my question is this how do i rotate the area between the functions $f(x)=x^2+1$, and $f(x)=2x^2-2$, whereby $x>0$ and $y>0$ and the region is bounded by the y and x axis.

This is a plot of the area I'm talking about.

Revolving the area around the y-axis is suppose to produce a vase, although I picture something of a shallow bowl.

Any help would be much appreciated. As I'm still quite new to Mathematica, an explanation of any code posted would be helpful.

• Please include the code you are trying in your question Commented May 8, 2015 at 0:30
• RevolutionPlot3D[{x^2 + 1, 2 x^2 - 2}, {x, 0, Sqrt[3]}, PlotRange -> All, RevolutionAxis -> "Y"] this is the code I'm trying but its not giving me what i know it should look like Commented May 8, 2015 at 0:33
• RevolutionPlot3D[{fx,fz},{t,t0,t1}] generates a plot of the surface obtained by rotating the parametric curve with x,z coordinates {fx, fz} around the z axis. So your input created a very different curve from what Plot shows. Check the documentation very carefully. Commented May 8, 2015 at 14:32

I think the plot will look more like what I believe the OP is expecting if a few options are added.

Show[RevolutionPlot3D[#, {x, 0, Sqrt[3]},
PlotRange -> {Automatic, Automatic, {0., Automatic}},
MeshStyle -> None,
Axes -> None,
Boxed -> False,
PlotStyle -> Opacity[.5]] & /@
{2 x^2 - 2, x^2 + 1, Piecewise[{{0, 0 < x < 1}}, Null]}]


Also note that I have put a bottom on the bowl.

Show[RevolutionPlot3D[#, {x, 0, Sqrt[3]}, MeshStyle -> None,
PlotStyle -> Opacity[.5]] & /@ {2 x^2 - 2,  x^2 + 1}]


• shouldn't it have a flat bottom as the domain of this pair of functions is restricted to x>0 and y>0 ? Commented May 8, 2015 at 1:52
• @errymerry I believe you can work that out easily by yourself ;) Commented May 8, 2015 at 3:34

Just for something different:

pp = ParametricPlot3D[{Sqrt[3] Cos[t], Sqrt[3] Sin[t], 4}, {t, 0,
2 Pi}, PlotStyle -> {Red, Thickness[0.04]}, Boxed -> False];
rp = RegionPlot3D[
0 < x^2 + y^2 < 3 &&
Max[0, 2 x^2 + 2 y^2 - 2] < z < x^2 + y^2 + 1, {x, -2, 2}, {y, -2,
2}, {z, 0, 5}, Mesh -> None, PlotStyle -> Red, PlotPoints -> 200,
PerformanceGoal -> "Quality", Boxed -> False];
Show[pp, rp, Axes -> False, Background -> Black]


Top ring just to deal with boundary of intersection...